Theorem txcmpb 21447

Description: The topological product of two nonempty topologies is compact iff the component topologies are both compact. (Contributed by Mario Carneiro, 14-Sep-2014.)

Hypotheses

Ref	Expression
txcmpb.1	|- X = U. R
txcmpb.2	|- Y = U. S

Assertion

Ref	Expression
txcmpb	|- ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( ( R tX S ) e. Comp <-> ( R e. Comp /\ S e. Comp ) ) )

Proof of Theorem txcmpb

Step	Hyp	Ref	Expression
1		simpr 477	. . . . 5 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( R tX S ) e. Comp )
2		simplrr 801	. . . . . . 7 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> Y =/= (/) )
3		fo1stres 7192	. . . . . . 7 |- ( Y =/= (/) -> ( 1st |` ( X X. Y ) ) : ( X X. Y ) -onto-> X )
4	2, 3	syl 17	. . . . . 6 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( 1st |` ( X X. Y ) ) : ( X X. Y ) -onto-> X )
5		txcmpb.1	. . . . . . . . 9 |- X = U. R
6		txcmpb.2	. . . . . . . . 9 |- Y = U. S
7	5, 6	txuni 21395	. . . . . . . 8 |- ( ( R e. Top /\ S e. Top ) -> ( X X. Y ) = U. ( R tX S ) )
8	7	ad2antrr 762	. . . . . . 7 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( X X. Y ) = U. ( R tX S ) )
9		foeq2 6112	. . . . . . 7 |- ( ( X X. Y ) = U. ( R tX S ) -> ( ( 1st |` ( X X. Y ) ) : ( X X. Y ) -onto-> X <-> ( 1st |` ( X X. Y ) ) : U. ( R tX S ) -onto-> X ) )
10	8, 9	syl 17	. . . . . 6 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( ( 1st |` ( X X. Y ) ) : ( X X. Y ) -onto-> X <-> ( 1st |` ( X X. Y ) ) : U. ( R tX S ) -onto-> X ) )
11	4, 10	mpbid 222	. . . . 5 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( 1st |` ( X X. Y ) ) : U. ( R tX S ) -onto-> X )
12	5	toptopon 20722	. . . . . . 7 |- ( R e. Top <-> R e. (TopOn ` X ) )
13	6	toptopon 20722	. . . . . . 7 |- ( S e. Top <-> S e. (TopOn ` Y ) )
14		tx1cn 21412	. . . . . . 7 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) )
15	12, 13, 14	syl2anb 496	. . . . . 6 |- ( ( R e. Top /\ S e. Top ) -> ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) )
16	15	ad2antrr 762	. . . . 5 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) )
17	5	cncmp 21195	. . . . 5 |- ( ( ( R tX S ) e. Comp /\ ( 1st |` ( X X. Y ) ) : U. ( R tX S ) -onto-> X /\ ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) ) -> R e. Comp )
18	1, 11, 16, 17	syl3anc 1326	. . . 4 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> R e. Comp )
19		simplrl 800	. . . . . . 7 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> X =/= (/) )
20		fo2ndres 7193	. . . . . . 7 |- ( X =/= (/) -> ( 2nd |` ( X X. Y ) ) : ( X X. Y ) -onto-> Y )
21	19, 20	syl 17	. . . . . 6 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( 2nd |` ( X X. Y ) ) : ( X X. Y ) -onto-> Y )
22		foeq2 6112	. . . . . . 7 |- ( ( X X. Y ) = U. ( R tX S ) -> ( ( 2nd |` ( X X. Y ) ) : ( X X. Y ) -onto-> Y <-> ( 2nd |` ( X X. Y ) ) : U. ( R tX S ) -onto-> Y ) )
23	8, 22	syl 17	. . . . . 6 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( ( 2nd |` ( X X. Y ) ) : ( X X. Y ) -onto-> Y <-> ( 2nd |` ( X X. Y ) ) : U. ( R tX S ) -onto-> Y ) )
24	21, 23	mpbid 222	. . . . 5 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( 2nd |` ( X X. Y ) ) : U. ( R tX S ) -onto-> Y )
25		tx2cn 21413	. . . . . . 7 |- ( ( R e. (TopOn ` X ) /\ S e. (TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) )
26	12, 13, 25	syl2anb 496	. . . . . 6 |- ( ( R e. Top /\ S e. Top ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) )
27	26	ad2antrr 762	. . . . 5 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) )
28	6	cncmp 21195	. . . . 5 |- ( ( ( R tX S ) e. Comp /\ ( 2nd |` ( X X. Y ) ) : U. ( R tX S ) -onto-> Y /\ ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) ) -> S e. Comp )
29	1, 24, 27, 28	syl3anc 1326	. . . 4 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> S e. Comp )
30	18, 29	jca 554	. . 3 |- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( R e. Comp /\ S e. Comp ) )
31	30	ex 450	. 2 |- ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( ( R tX S ) e. Comp -> ( R e. Comp /\ S e. Comp ) ) )
32		txcmp 21446	. 2 |- ( ( R e. Comp /\ S e. Comp ) -> ( R tX S ) e. Comp )
33	31, 32	impbid1 215	1 |- ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( ( R tX S ) e. Comp <-> ( R e. Comp /\ S e. Comp ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   U.cuni 4436   X. cxp 5112   |` cres 5116   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   Compccmp 21189   tX ctx 21363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cmp 21190  df-tx 21365

This theorem is referenced by: (None)